LTC7860 limits current if sense threshold breached?

I think that the LTC7860 will limit the current. There doesn't seem to be anything in the data sheet to say that the protection only applies in an over-voltage condition. The current limit will protect the output wiring, the inductor, the MOSFET and the sensing resistor, so I can't see why it would ever be disabled.

The data sheet says:- "An inductor current limit condition is detected when VSENSE exceeds 95mV. When the current limit threshold is exceeded, the P-channel MOSFET is immediately turned off by pulling the GATE voltage to VIN regardless of the controller input."

The 600 W power module will only take 35 A at full load with an 18 V supply. Any less load or higher supply voltage will result in less current being taken.

The design appears to be marginal for 18 V / 600 W load, but it could be that those conditions don't need to be met. If the load is less than the power module's maximum rating, 35 A won't ever be consumed.

You could try making a circuit with a much lower rating. Just use a 3 Ohm resistor and the current limit will be 35 mA and it will be much easier to test the overload conditions.